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A306387 Number of partitions of sigma_1(n) into divisors of n. 0
1, 2, 2, 6, 2, 27, 2, 26, 7, 31, 2, 574, 2, 38, 33, 166, 2, 879, 2, 924, 39, 52, 2, 23732, 9, 59, 47, 1403, 2, 34256, 2, 1626, 55, 73, 47, 230819, 2, 80, 61, 50888, 2, 65638, 2, 2709, 1734, 94, 2, 2117920, 11, 3038, 77, 3536, 2, 113448, 65, 97298, 83, 115, 2, 19613170, 2, 122, 2601, 25510, 73, 180350 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equality sigma_1(n) = Sum{d|n} d defines a partition of sigma_1(n) with divisors of n with distinct terms. The proposed sequence lists the partitions of sigma_1(n) with n's divisors, not necessarily distinct.

For n prime number, sigma_1(n) = n+1 and there are only two partitions: n and 1+1+1+...+1, 1 n times.

LINKS

Table of n, a(n) for n=1..66.

EXAMPLE

For n = 4, sigma_1(4) = 7, Divisors(4) = {1,2,4} and 7 = 4+2+1 = 4+1+1+1 = 2+2+2+1 = 2+2+1+1+1 = 2+1+1+1+1+1 = 1+1+1+1+1+1+1.

For n = 9, sigma_1(9) = 13, Divisors(9) = {1,3,9} and 13 = 9+3+1 = 9+1+1+1+1 = 3+3+3+3+1 = 3+3+3+1+1+1+1 = 3+3+1+1+1+1+1+1+1 = 3+1+1+1+1+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1+1+1.

PROG

(MAGMA) v:=[1..47];

for u in v do

u, #RestrictedPartitions(SumOfDivisors(u), {d:d in Divisors(u)});

end for;

(MAGMA)

a:= func< n | #RestrictedPartitions(SumOfDivisors(n), {d:d in Divisors(n)}) >; [ a(n) : n in [1..47] ];

(PARI) numbpartUsing(n, v, mx=#v)=if(n<1, return(n==0)); sum(i=1, mx, numbpartUsing(n-v[i], v, i)) \\ inefficient;

a(n) = numbpartUsing(sigma(n), divisors(n)); \\ after A018818; Michel Marcus, Feb 27 2019

CROSSREFS

Cf, A000005, A000041, A000203, A018818.

Sequence in context: A284839 A286376 A100346 * A308692 A319352 A300834

Adjacent sequences:  A306384 A306385 A306386 * A306388 A306389 A306390

KEYWORD

nonn

AUTHOR

Marius A. Burtea, Feb 26 2019

STATUS

approved

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Last modified February 24 08:38 EST 2020. Contains 332203 sequences. (Running on oeis4.)